Alphabeta MathAlgebraic Differentials Separability and Smooth Local Presentations — Examples
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Fields and Cyclotomic Extensions
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
Algebraic Geometry
Commutative Algebra
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Integral Extensions and Going Up
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
Linear Algebra
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Example
Differentials of a polynomial ring and of a cuspidal hypersurface
Counterexample
An arbitrary algebra map does not make differentials a base change
Example
A finite separable extension and an inseparable extension with differentials
Example
A cuspidal plane curve is standard smooth away from the cusp
Counterexample
Regular field factors can have a nonregular tensor product
Example
Geometric parameters of a projection of affine spaces
Sources
- Vakil §22.2.7, pp.575–577
- Stacks Algebra 10.131.9–10 and 10.131.14
- Stacks Algebra 10.131.8 and 10.131.12
- Vakil §22.2.3, p.575
- Vakil §22.2.F, p.577
- Stacks Algebra 10.131.14 and 10.143.2
- Stacks Algebra 10.137.5 (tag 00T6)
- Stacks Algebra 10.166.1–2 (tags 0381, 0382)
- Stacks Algebra 10.137.5 (tag 00T6) and 10.140.5 (tag 00TV)
- Vakil §26.2.F, pp.690–693